MORPHOLOGY AND MATHEMATICS. 895 



the second place, the " facial axis," for instance, as ordinarily used in the anthro- 

 pological comparison of one human skull with another, or of the human skull with 

 the gorilla's, is in all cases treated as a straight line ; but our investigation has shown 

 that rectilinear axes only meet the case in the simplest and most closely related 

 transformations ; and that, for instance, in the anthropoid skull no rectilinear axis is 

 homologous with a rectilinear axis in a man's skull, but what is a straight line in the 

 one has become a certain definite curve in the other. 



As a final illustration I have drawn the outline of a Dog's skull (fig. 66), and 

 inscribed it in a network comparable with the Cartesian network of the human skull 

 in fig. 62. Here we attempt to bridge over a wider gulf than we have crossed in 

 any of our former comparisons. But, nevertheless, it is obvious that our method 

 still holds good, in spite of the fact that there are various specific differences, such 

 as the open or closed orbit, etc., which have to be separately described and accounted 

 for. We see that the chief essential differences in plan between the dog's skull and 



Fir;. 66. — Skull of dog, compared with the human skull of fig. 62. 



the man's lie in the fact that, relatively speaking, the former tapers away in front, 

 a triangular taking the place of a rectangular conformation ; secondly, that, coincident 

 with the tapering off, there is a progressive elongation, or pulling out, of the whole 

 forepart of the skull ; and lastly, as a minor difference, that the straight vertical 

 ordinates of the human skull become curved, with their convexity directed forwards, 

 in the dog. While the net result is that in the dog, just as in the chimpanzee, 

 the brain-pan is smaller and the jaws are larger than in man, it is interesting and 

 important to observe that the co-ordinate network of the ape is by no means 

 intermediate between those which fit the other two. The mode of deformation is 

 on different lines ; and, while it may be correct to say that the chimpanzee and 

 the baboon are more brute-like, it would be by no means accurate to assert that 

 they are more doglike, than man. 



In this paper I have dealt with plane co-ordinates only, and have made no 

 mention of the more difficult subject of systems of co-ordinates in three-dimensional 

 space. But it is obvious that, if the difficulties of description and representation 

 could be overcome, it is by means of such co-ordinates in space that we should obtain 

 an adequate and satisfying picture of the processes of deformation and the directions 

 of growth. 



